A380441 Sum of the nonprimes dividing n and the number of distinct primes dividing n.

1, 2, 2, 6, 2, 9, 2, 14, 11, 13, 2, 25, 2, 17, 18, 30, 2, 36, 2, 37, 24, 25, 2, 57, 27, 29, 38, 49, 2, 65, 2, 62, 36, 37, 38, 88, 2, 41, 42, 85, 2, 87, 2, 73, 72, 49, 2, 121, 51, 88, 54, 85, 2, 117, 58, 113, 60, 61, 2, 161, 2, 65, 96, 126, 68, 131, 2, 109, 72, 133, 2, 192, 2, 77, 118, 121, 80, 153, 2, 181, 119, 85, 2, 215, 88, 89, 90, 169, 2, 227, 94, 145, 96

Offset

1,2

Comments

Inverse Möbius transform of A005451(n).

For each divisor d of n, add 1 if d is prime, else add d.

Links

Table of n, a(n) for n=1..93.

Formula

a(n) = sigma(n) - sopf(n) + omega(n).

a(n) = Sum_{d|n} d^c(d), where c = A005171.

a(n) = Sum_{d|n} A005451(d).

a(p^k) = 1 - p + (p^(k+1)-1)/(p-1) for p prime, k >= 1. - Wesley Ivan Hurt, Jul 02 2025

a(n) = A023890(n) + A001221(n). - Wesley Ivan Hurt, Aug 31 2025

Mathematica

Table[DivisorSigma[1, n] - Sum[p, {p, Select[Divisors[n], PrimeQ]}] + PrimeNu[n], {n, 100}]

Crossrefs

Cf. A000203 (sigma), A001221 (omega), A005171 (char nonprimes), A005451, A008472 (sopf), A023890.

Sequence in context: A096869 A345315 A385076 * A154009 A297792 A266722

Adjacent sequences: A380438 A380439 A380440 * A380442 A380443 A380444

Keyword

nonn,easy

Author

Wesley Ivan Hurt, Jun 21 2025

Status

approved